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46
Lower Bounds For The Polynomial Calculus
, 1998
"... We show that polynomial calculus proofs (sometimes also called Groebner proofs) of the pigeonhole principle PHP n must have degree at least (n=2)+1 over any field. This is the first nontrivial lower bound on the degree of polynomial calculus proofs obtained without using unproved complexity assumpt ..."
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Cited by 56 (6 self)
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We show that polynomial calculus proofs (sometimes also called Groebner proofs) of the pigeonhole principle PHP n must have degree at least (n=2)+1 over any field. This is the first nontrivial lower bound on the degree of polynomial calculus proofs obtained without using unproved complexity assumptions. We also show that for some modifications of PHP n , expressible by polynomials of at most logarithmic degree, our bound can be improved to linear in the number of variables. Finally, we show that for any Boolean function f n in n variables, every polynomial calculus proof of the statement "f n cannot be computed by any circuit of size t," must have degree t=n). Loosely speaking, this means that low degree polynomial calculus proofs do not prove NP 6 P=poly.
Unprovability of Lower Bounds on the Circuit Size in Certain Fragments of Bounded Arithmetic
 IN IZVESTIYA OF THE RUSSIAN ACADEMY OF SCIENCE, MATHEMATICS
, 1995
"... We show that if strong pseudorandom generators exist then the statement “α encodes a circuit of size n (log ∗ n) for SATISFIABILITY ” is not refutable in S2 2 (α). For refutation in S1 2 (α), this is proven under the weaker assumption of the existence of generators secure against the attack by smal ..."
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Cited by 55 (6 self)
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We show that if strong pseudorandom generators exist then the statement “α encodes a circuit of size n (log ∗ n) for SATISFIABILITY ” is not refutable in S2 2 (α). For refutation in S1 2 (α), this is proven under the weaker assumption of the existence of generators secure against the attack by small depth circuits, and for another system which is strong enough to prove exponential lower bounds for constantdepth circuits, this is shown without using any unproven hardness assumptions. These results can be also viewed as direct corollaries of interpolationlike theorems for certain “split versions” of classical systems of Bounded Arithmetic introduced in this paper.
Pseudorandom Generators Hard for kDNF Resolution and Polynomial Calculus Resolution
, 2003
"... A pseudorandom generator G n : f0; 1g is hard for a propositional proof system P if (roughly speaking) P can not ef ciently prove the statement G n (x 1 ; : : : ; x n ) 6= b for any string b 2 . We present a function (m 2 ) generator which is hard for Res( log n); here Res(k) is the ..."
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Cited by 50 (4 self)
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A pseudorandom generator G n : f0; 1g is hard for a propositional proof system P if (roughly speaking) P can not ef ciently prove the statement G n (x 1 ; : : : ; x n ) 6= b for any string b 2 . We present a function (m 2 ) generator which is hard for Res( log n); here Res(k) is the propositional proof system that extends Resolution by allowing kDNFs instead of clauses.
On provably disjoint NPpairs
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY
, 1994
"... In this paper we study the pairs (U; V ) of disjoint NPsets representable in a theory T of Bounded Arithmetic in the sense that T proves U " V = ;. For a large variety of theories T we exhibit a natural disjoint NPpair which is complete for the class of disjoint NPpairs representable in T ..."
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Cited by 42 (2 self)
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In this paper we study the pairs (U; V ) of disjoint NPsets representable in a theory T of Bounded Arithmetic in the sense that T proves U " V = ;. For a large variety of theories T we exhibit a natural disjoint NPpair which is complete for the class of disjoint NPpairs representable in T . This allows us to clarify the approach to showing independence of central open questions in Boolean complexity from theories of Bounded Arithmetic initiated in [11]. Namely, in order to prove the independence result from a theory T , it is sufficient to separate the corresponding complete NPpair by a (quasi)polytime computable set. We remark that such a separation is obvious for the theory S(S 2 ) + S \Sigma 2 \Gamma PIND considered in [11], and this gives an alternative proof of the main result from that paper.
A New Proof of the Weak Pigeonhole Principle
, 2000
"... The exact complexity of the weak pigeonhole principle is an old and fundamental problem in proof complexity. Using a diagonalization argument, Paris, Wilkie and Woods [16] showed how to prove the weak pigeonhole principle with boundeddepth, quasipolynomialsize proofs. Their argument was further re ..."
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Cited by 35 (2 self)
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The exact complexity of the weak pigeonhole principle is an old and fundamental problem in proof complexity. Using a diagonalization argument, Paris, Wilkie and Woods [16] showed how to prove the weak pigeonhole principle with boundeddepth, quasipolynomialsize proofs. Their argument was further refined by Kraj'icek [9]. In this paper, we present a new proof: we show that the the weak pigeonhole principle has quasipolynomialsize LK proofs where every formula consists of a single AND/OR of polylog fanin. Our proof is conceptually simpler than previous arguments, and is optimal with respect to depth. 1 Introduction The pigeonhole principle is a fundamental axiom of mathematics, stating that there is no onetoone mapping from m pigeons to n holes when m ? n. It expresses Department of Mathematics and Computer Science, Clarkson University, Potsdam, NY 136995815, U.S.A. alexis@clarkson.edu. Research supported by NSF grant CCR9877150. y Department of Computer Science, University o...
Theories for Complexity Classes and their Propositional Translations
 Complexity of computations and proofs
, 2004
"... We present in a uniform manner simple twosorted theories corresponding to each of eight complexity classes between AC and P. We present simple translations between these theories and systems of the quanti ed propositional calculus. ..."
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Cited by 33 (7 self)
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We present in a uniform manner simple twosorted theories corresponding to each of eight complexity classes between AC and P. We present simple translations between these theories and systems of the quanti ed propositional calculus.
Elusive functions and lower bounds for arithmetic circuits
 Electronic Colloquium in Computational Complexity
, 2007
"... A basic fact in linear algebra is that the image of the curve f(x) = (x1, x2, x3,..., xm), say over C, is not contained in any m − 1 dimensional affine subspace of Cm. In other words, the image of f is not contained in the image of any polynomialmapping1 Γ: Cm−1 → Cm of degree 1 (that is, an affin ..."
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Cited by 22 (2 self)
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A basic fact in linear algebra is that the image of the curve f(x) = (x1, x2, x3,..., xm), say over C, is not contained in any m − 1 dimensional affine subspace of Cm. In other words, the image of f is not contained in the image of any polynomialmapping1 Γ: Cm−1 → Cm of degree 1 (that is, an affine mapping). Can one give an explicit example for a polynomial curve f: C → Cm, such that, the image of f is not contained in the image of any polynomialmapping Γ: Cm−1 → Cm of degree 2? In this paper, we show that problems of this type are closely related to proving lower bounds for the size of general arithmetic circuits. For example, any explicit f as above (with the right notion of explicitness2), of degree up to 2mo(1) , implies superpolynomial lower bounds for computing the permanent over C. More generally, we say that a polynomialmapping f: Fn → Fm is (s, r)elusive, if for every polynomialmapping Γ: Fs → Fm of degree r, Image(f) � ⊂ Image(Γ).
Structure and Definability in General Bounded Arithmetic Theories
, 1999
"... This paper is motivated by the questions: what are the \Sigma ..."
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Cited by 21 (6 self)
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This paper is motivated by the questions: what are the \Sigma
Tautologies From PseudoRandom Generators
, 2001
"... We consider tautologies formed from a pseudorandom number generator, dened in Krajcek [12] and in Alekhnovich et.al. [2]. We explain a strategy of proving their hardness for EF via a conjecture about bounded arithmetic formulated in Krajcek [12]. Further we give a purely nitary statement, in a ..."
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Cited by 19 (4 self)
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We consider tautologies formed from a pseudorandom number generator, dened in Krajcek [12] and in Alekhnovich et.al. [2]. We explain a strategy of proving their hardness for EF via a conjecture about bounded arithmetic formulated in Krajcek [12]. Further we give a purely nitary statement, in a form of a hardness condition posed on a function, equivalent to the conjecture. This is accompanied by a brief explanation, aimed at nonlogicians, of the relation between propositional proof complexity and bounded arithmetic. It is a fundamental problem of mathematical logic to decide if tautologies can be inferred in propositional calculus in substantially fewer steps than it takes to check all possible truth assignments. This is closely related to the famous P/NP problem of Cook [3]. By propositional calculus I mean any textbook system based on a nite number of inference rules and axiom schemes that is sound and complete. The qualication substantially less means that the nu...
A Satisfiability Algorithm for AC 0
, 2011
"... We consider the problem of efficiently enumerating the satisfying assignments to AC 0 circuits. We give a zeroerror randomized algorithm which takes an AC 0 circuit as input and constructs a set of restrictions which partitions {0, 1} n so that under each restriction the value of the circuit is cons ..."
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Cited by 17 (2 self)
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We consider the problem of efficiently enumerating the satisfying assignments to AC 0 circuits. We give a zeroerror randomized algorithm which takes an AC 0 circuit as input and constructs a set of restrictions which partitions {0, 1} n so that under each restriction the value of the circuit is constant. Let d denote the depth of the circuit and cn denote the number of gates. This algorithm runs in time C2 n(1−µc,d) where C  is the size of the circuit for µc,d ≥ 1/O[lg c + d lg d] d−1 with probability at least 1 − 2 −n. As a result, we get improved exponential time algorithms for AC 0 circuit satisfiability and for counting solutions. In addition, we get an improved bound on the correlation of AC 0 circuits with parity. As an important component of our analysis, we extend the H˚astad Switching Lemma to handle multiple kcnfs and kdnfs. 1